compound
functions - annuities

The
most common valuation problem is the valuation of future periodic
income from investment properties, usually rent. In part 10 the
present value of rental income for the period 0 to perpetuity was
found with the capitalization in perpetuity formula:

MV
= NAI * 100/CR

Where:

MV
= market value

NAI
= net annual income

CR.
= capitalization rate as a %.

In
this part we are concerned with the future and present values of
periodic level payments (that is, the same amount each period)
at periods less than perpetuity. Periodic means that the
payments are made at regular periods for example; per annum.

The
formulae covered in this part cannot be used for expected incomes
that are uneven in time or amount. For complex cash flows discounted
cash flow is used.

YEARS
PURCHASE (YP)

The
factor; 100/CR in the above capitalization formula is known as the
years purchase (YP) and is the maximum factor or multiplier by
which rent can be multiplied to obtain market value.

EXAMPLE

A
capitalization rate of 8% pa has a years purchase of 100/8 =12.5.

Years
purchase is commonly used in business valuations and can be equated
with the break even point or payback period used in feasibility
studies.

For
valuation purposes the capitalization rate expressed as a percentage
is the most commonly used measure of return on investment. However,
it cannot be compared directly with opportunity cost investments such
as bank interest rates and the return on government bonds, as it does
not take into account an important part of real estate investment
return; capital gains.

FUTURE
VALUE OF 1 PER ANNUM (FVPMT)

For
example, to determine the future value of 1 pa over 20 years at 12%
pa:

STEP
1:

Determine
the base: b = 1+i = 1.12

STEP
2:

Determine
the future value factor (FV):

FV = bn =
1.1220 = 9.646

STEP
3:

The
future value of 1 per annum (FVPMT) can be found with the following
formula:

FVPMT
= (FV 1)/(i/l00)

Where:

FVPMT = future value
of 1 pa factor

FV
= future value of 1 = interest rate as a % eg 5% is 5

Therefore,
the FVPMT for the expected cash flow above:

FVPMT
= (9.646 1)/ 0.12 = 72.05

STEP
4:

Multiply
the FVPMT factor by the periodic income for example, $5 000 pa:

FVPMT(5000) = 72.05
* $5 000 = $360 250

The
FVPMT factor shows that 1 per annum will accumulate to 72.052 if
invested over 20 years at 12%pa. This is shown on the following time
scale:

1
pa @ 12% = 72. 052

=====================================>

0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

years

If
the net rental income for the above property is $200 000 per annum
the future amount is:

FVPMT(200
000)= 72.052 * 200 000 = 14 410 500

say
$14 400 000

PRESENT
VALUE OF 1 PER ANNUM (PVPMT)

The
most useful compound formula in valuation practice is the present
value of a future periodic level cash flow. For example, the present
value of rents due under a lease agreement.

EXAMPLE

The
expected net rental income from a 10 year lease is expected to be

$10
000pa.

Determine
the present value of $10 000 pa for 10 years at 9 % pa.

STEP
1:

Determine
the base: b= 1+i = 1+0.09 = 1.09

STEP
2:

Determine
FV: FV = 1.0910 = 2.367

STEP
3

Determine
PV: PV = 1/FV = 1/2.367 = 0.4224

STEP
4:

PVPMT
is found with the following formula: PVPMT = (1 - PV)/i

Therefore,
for the above example:

= (1 - 0.4224)/0.09 =
6.418

STEP
5:

Multiply
PVPMT by the annual income:

PVPMT(10000)
= 6.418 * 10000 = $64180 say
$64 200

Therefore,
the present value of a future rental stream of $10 000 over 10 years
at 9% pa = $64 200.

REPLACEMENT
OF CAPITAL IN REAL ESTATE INVESTMENTS

For
a normal investment rate of return it is assumed that the capital
invested is being replaced at the same rate of interest as that for
the return on investment:

LEASEBACK
EXAMPLE

A
new factory premises is built for a company and then leased back to
that company for the life of the building.

The
required return is 12% and the money invested by the financier in the
leaseback is $100 000. Therefore, the required annual return on
investment should be $100 000 * 12% = $12 000.

However,
using the compound formula, the actual return on investment is $13
389 that is, an extra $1389 pa.

Why
the extra amount?

Before
we answer, which of the following investments would you prefer,
assuming that the risks applicable to both are the same:

the leaseback agreement as outlined above; or

investing in a bank account that returns 12% per
annum?

ANSWER

The
bank account!

At
the end of 20 years the investor will have his/her original capital
of $100 000 intact whereas under the leaseback agreement the building
will have NIL value in 20 years time. Therefore, investors in
depreciating assets such as buildings, must allow for the replacement
of capital over the life of the investment.

REPLACEMENT
OF CAPITAL - THE SINKING FUND

Investors
in buildings allow for the replacement of capital over the life of
the building by way of a sinking fund. The sinking fund is a special
fund set aside with a periodic payment out of the rents, sufficient
to amount to the value of the building over its expected life.
Therefore, at the end of that period, the building investor will have
sufficient funds to rebuild the building and start all over again.

EXAMPLE:
BODY COPORATE OF A STRATA (UNIT) SCHEME

A
typical sinking fund problem is the statutory requirement of the body
corporate (owner’s association) of a strata (unit) title plan to
replace for example, carpets in the common property at the end of
their life. The carpets have a life expectancy of 15 years and are
expected to cost

$200
000 to replace. The relevant bank account is paying 12%pa.

The
body corporate has asked you to determine how much their annual
contribution to a sinking fund should be.

ANSWER

The
sinking fund factor (SFF) is that annual payment sufficient to
amount to $1 over the life of the depreciating asset to be replaced.
Therefore, it is the reciprocal of the FVPMT factor:

SINKING
FUND FACTOR

SFF = 1/FVPMT

Where:

SFF = sinking fund factor

FVPMT = the future
value of 1pa

To
calculate the necessary contributions to the body corporate's
(owners’ corporation) fund:

STEP
1

Calculate
FVPMT:

base = 1.12

FV = 1.1215 =
5.474

FVPMT = (FV1)/(i/100)

= (5.474)/0.12 =
37.28

STEP
2

Determine
the sinking fund factor (SFF):

SFF = 1/37.28 =
`0.0268

STEP
3

Determine
the sinking fund contribution:

200
000 * 0.0268 = $5 360pa

That
is, $5 360 pa will amount to about $200 000 (error due to rounding)
over 15 years at 12%pa.

SINGLE
RATE ASSUMPTION

In
the leaseback example above, the question was raised about the
apparent surplus when the company paid a rent of $13 389 pa. The
required rate of return was $12 000 pa (12%) leaving a residue of
$1389pa to replace the building by way of a sinking fund:

FVPMT(1) = 72.05

FVPMT(1388.67) =
72.05 * $1389

= $100 077

say $100 000

(difference
due to rounding)

Therefore,
a single rate factor includes a sinking fund component replacing the
investment amount at the SAME rate as the required return on
investment. In the above example; 12%pa.